The kinetics of linear systems with special reference to periodic reactions

Hearon, John Z.

doi:10.1007/bf02476377

Cited by 138 publications

(44 citation statements)

References 18 publications

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“…We start with: This theorem has an interesting history. Parter pointed out in [3] that AEQ of semi positive type is symmetrizable, basing his result upon previous work by Hearon [6] and Goldberg [4 , 5]. Independently the present author pointed out in [1] that A is symmetrizable and, apparently, first noted that this co uld be accomplished with a diagonal matrix.…”

Section: Semi~ositive Elements Of F2supporting

confidence: 69%

Matrices of class J2

Maybee¹

1967

J. RES. NATL. BUR. STAN. SECT. B. MATH. MATH. PHYS.

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Section: Semi~ositive Elements Of F2supporting

confidence: 69%

Matrices of class J2

Maybee¹

1967

J. RES. NATL. BUR. STAN. SECT. B. MATH. MATH. PHYS.

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“…The possibility of oscillations in cyclical chains of chemical reactions was investigated by Hearon (1953) who showed the absence of sustained oscillations in such systems in the case of linear kinetics. The cyclical nature of some biochemical pathways does not ensure per se the occurrence of metabolic oscillations.…”

Section: Dominoes Linked To An Independent Clockmentioning

confidence: 99%

A Model for a Network of Phosphorylation–dephosphorylation Cycles Displaying the Dynamics of Dominoes and Clocks

Gonze¹,

Goldbeter²

2001

Journal of Theoretical Biology

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We consider a model for a network of phosphorylation}dephosphorylation cycles coupled through forward and backward regulatory interactions, such that a protein phosphorylated in a given cycle activates the phosphorylation of a protein by a kinase in the next cycle as well as the dephosphorylation of a protein by a phosphatase in a preceding cycle. The network is cyclically organized in such a way that the protein phosphorylated in the last cycle activates the kinase in the "rst cycle. We study the dynamics of the network in the presence of both forward and backward coupling, in conditions where a threshold exists in each cycle in the amount of protein phosphorylated as a function of the ratio of kinase to phosphatase maximum rates. We show that this system can display sustained (limit-cycle) oscillations in which each cycle in the pathway is successively turned on and o!, in a sequence resembling the fall of a series of dominoes. The model thus provides an example of a biochemical system displaying the dynamics of dominoes and clocks (Murray & Kirschner, 1989). It also shows that a continuum of clock waveforms exists of which the fall of dominoes represents a limit. When the cycles in the network are linked through only forward (positive) coupling, bistability is observed, while in the presence of only backward (negative) coupling, the system can display multistability or oscillations, depending on the number of cycles in the network. Inhibition or activation of any kinase or phosphatase in the network immediately stops the oscillations by bringing the system into a stable steady state; oscillations resume when the initial value of the kinase or phosphatase rate is restored. The progression of the system on the limit cycle can thus be temporarily halted as long as an inhibitor is present, much as when a domino is held in place. These results suggest that the eukaryotic cell cycle, governed by a network of phosphorylation}dephosphorylation reactions in which the negative control of cyclin-dependent kinases plays a prominent role, behaves as a limit-cycle oscillator impeded in the presence of inhibitors. We contrast the case where the sequence of domino-like transitions constitutes the clock with the case where the sequence of transitions is passively coupled to a biochemical oscillator operating as an independent clock. Academic Press

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“…There are no analytical expressions for the eigenvalues of Eqn (93) but it is known that they are real and negative (cf. [94]). After a sufficiently long time, only the eigenvector belonging to the eigenvalue with the smallest absolute value (denoted, say, by &) determines the relaxation process.…”

Section: Transient Timesmentioning

confidence: 99%

Mathematical analysis of enzymic reaction systems using optimization principles

Schuster

Holzhütter

1991

European Journal of Biochemistry

131

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Contents Introduction Optimization of the catalytic properties of single enzymes Irreversible two‐step kinetic mechanism Reversible three‐step kinetic mechanism Cooperative enzymes Optimization of multienzyme systems Basic equations for unbranched reaction chains Maximization ofsteady‐state fluxes Transient times Sensitivity to external signals Implications of the solvent capacity and osmotic balance Metabolic channelling Thermodynamic principles Optimal stoichiometries Time hierarchy in biochemical networks Final remarks

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The kinetics of linear systems with special reference to periodic reactions

Cited by 138 publications

References 18 publications

Matrices of class J2

Matrices of class J2

A Model for a Network of Phosphorylation–dephosphorylation Cycles Displaying the Dynamics of Dominoes and Clocks

Mathematical analysis of enzymic reaction systems using optimization principles

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